Problems

The equations $$\begin{array}{}\text{(1)}& a{x}^2+bx+c=0\end{array}$$ and $$\begin{array}{}\text{(2)}& -a{x}^2+bx+c\end{array}$$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_3$ of the equation $\frac{1}{2}a{x}^2+bx+c$ such that either $x_1\le x_3\le x_2$ or $x_1\ge x_3\ge x_2$.

Prove that if $x_0^4+a_1{x}_0^3+a_2{x}_0^2+a_3{x}_0+a_4$ and $4x_0^3+3a_1{x}_0^2+2a_2{x}_0+a_3=0$ then $x^4+a_1{x}^3+a_2{x}^2+a_{3}x+a_4$ is divisible by $(x-x_0{)}^2$.

The segment $OA$ is given. From the end of the segment $A$ there are 5 segments $A{B}_1,A{B}_2,A{B}_3,A{B}_4,A{B}_5$. From each point $B_i$ there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment (except point $O$).

Solve the equation $x^3-\lfloor x\rfloor =3$.

There is a system of equations $$\begin{array}{rl}\ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0.\end{array}$$ Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.

There are two sets of numbers made up of 1s and $-1$s, and in each there are 2022 numbers. Prove that in some number of steps it is possible to turn the first set into the second one if for each step you are allowed to simultaneously change the sign of any 11 numbers of the starting set. (Two sets are considered the same if they have the same numbers in the same places.)

Two people play a game with the following rules: one of them guesses a set of integers $(x_1,x_2,\dots ,x_n)$ which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum $a_1{x}_1+\cdots +a_n{x}_n$, where $(a_1,\dots ,a_n)$ is any set. What is the smallest number of questions for which the guesser recognizes the intended set?

Given $n$ points that are connected by segments so that each point is connected to some other and there are no two points that would be connected in two different ways. Prove that the total number of segments is $n-1$.

A system of points connected by segments is called “connected” if from each point one can go to any other one along these segments. Is it possible to connect five points to a connected system so that when erasing any segment, exactly two connected points systems are formed that are not related to each other? (We assume that in the intersection of the segments, the transition from one of them to another is impossible).

Two players play on a square field of size $99\times 99$, which has been split onto cells of size $1\times 1$. The first player places a cross on the center of the field; After this, the second player can place a zero on any of the eight cells surrounding the cross of the first player. After that, the first puts a cross onto any cell of the field next to one of those already occupied, etc. The first player wins if he can put a cross on any corner cell. Prove that with any strategy of the second player the first can always win.